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Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

2
votes
1answer
33 views

Cardinal exponentiation without generalized continuum hypothesis

First I have to confess that I don't know about set theory language. Let $A$ and $B$ be infinite cardinals with $A>B$. My question is: $A^B=A$? (without assuming generalized continuum hypothesis) ...
0
votes
2answers
26 views

How does my bijection between the natural numbers and the powerset of natural numbers break down? [duplicate]

Lets consider some natural number x in binary. Let the least significant digit represent the inclusion or exclusion of 0, the next least significant represent 1, and so on upwards. For some examples: ...
0
votes
1answer
71 views

Measurable cardinals: non-trivial two-valued measures

while doing some exercises about measurable cardinals, I got stuck on this one: If $κ$ is the minimal cardinal that carries a non-trivial two-valued measure, then how can one prove that $κ$ is ...
1
vote
2answers
58 views

Let $A_{n}=\left\{ f \in \left\{ 0,1\right\}^{\mathbb N}: f(n)=0 \right\} $

Let $$A_{n}=\left\{ f \in \left\{ 0,1\right\}^{\mathbb N}: f(n)=0 \right\} $$ Find (a) $$| \bigcap_{m \in \mathbb N}^{} \bigcup_{n \ge m}^{} A_{n}|$$ (b) $$|\left\{ 0,1\right\}^{\mathbb N} \...
1
vote
2answers
29 views

If $A\sim B$(both dedekind infinite), is it then that $A\sim B\cup \{x\}$

If the symbol $A\sim B $signifies that there is a bijection between A and B, and We take our sets to be dedekind infinite, then is the following correct? If not, what is the counter example?:$$A \sim ...
1
vote
0answers
67 views

Proof of the Solovay Theorem in Jech

The Solovay Theorem says: Let $\kappa$ be a regular uncountable cardinal. Then every stationary subset is the disjoint union of $\kappa$-many stationary subsets. [Jech, Theorem 8.8, p. 95] One ...
1
vote
1answer
50 views

Finite chain condition - Variation of Martin's Axiom statement

In the following $k$ and $w$ will be cardinal numbers. Consider the classical statement $MA(k)$: For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family ...
10
votes
3answers
2k views

Questions about Aleph-Aleph-Null

Note: I apologize in advance for not using proper notation on some of these values, but this is literally my first post on this site and I do not know how to display these values correctly. I ...
2
votes
1answer
56 views

If $\bigcup_{i=1}^\infty A_i$ has cardinality $\kappa$, then some $A_i$ has cardinality $\kappa$?

Is the following claim true? If $\bigcup_{i=1}^\infty A_i$ has cardinality $\kappa$, then some $A_i$ has cardinality $\kappa$. Here $\kappa$ is uncountable. I'm interested in the particular case $\...
0
votes
1answer
31 views

A question about the proof about strongly inaccessible cardinal

My textbook Introduction to Set Theory by Hrbacek and Jech presents Theorem 3.13 and its corresponding proof as follows: Since the authors refer to Theorem 2.2, I post it here for reference: ...
1
vote
1answer
26 views

Commutative Noetherian ring with distinct ideals having distinct index

Let $R$ be a Noetherian commutative, infinite ring with unity such that distinct ideals have distinct index i.e. if $I,J$ are ideals of $R$ and $I \ne J$ , then $R/I$ and $R/J$ are not bijective as ...
0
votes
1answer
42 views

A question about a proof of Hausdorff's Formula

My textbook Introduction to Set Theory by Hrbacek and Jech presents Hausdorff's Formula: and its corresponding proof: I am unable to deduce 3. from 1. and 2. as stated in the proof. Each ...
0
votes
0answers
8 views

Counting equivalence classes. [duplicate]

Let $X$ denote the set of real transcendental numbers. Define the relation $\sim$ on $X$ by $x\sim y $ iff $x-y \in \mathbb{Q}$. Let $Y$ denote the set of equivalence classes generated by $\sim$ ...
0
votes
0answers
48 views

Is my understanding of this proof about cardinality correct?

In my textbook Introduction to Set Theory by Hrbacek and Jech, there is a theorem: and its corresponding proof: I would like to ask if my understanding of the proof in case $\color{blue}{\...
0
votes
1answer
45 views

Is my understanding of a proof from textbook Introduction to Set Theory by Hrbacek and Jech correct?

3.8 Therorem Let us assume the Generalized Continuum Hypothesis. If $\aleph_\alpha$ is a regular cardinal, then $$\aleph_\alpha^{\aleph_\beta}=\begin{cases} \aleph_\alpha&\text{if }\beta<\alpha\...
0
votes
1answer
22 views

Is every (possibly infinite) sum of cardinal numbers defined?

Hrbacek and Jech gives the following definition of cardinal addition: My question is: given an indexed system of cardinals $\left \langle \kappa_{i} |i\in I \right \rangle$ does there exist a system $...
0
votes
1answer
63 views

Let $\aleph_\alpha$ be a singular cardinal and $2^{\aleph_\xi}=\aleph_\beta$ for all $\xi<\alpha$. Then $2^{\aleph_\alpha}=\aleph_\beta$

Let $\aleph_\alpha$ be a singular cardinal. Let us assume that the value of $2^{\aleph_\xi}$ is the same for all $\xi<\alpha$, say $2^{\aleph_\xi}=\aleph_\beta$. Then $2^{\aleph_\alpha}=\aleph_\...
1
vote
1answer
65 views

A question regarding Brian M. Scott's proof that $\text{cf}(\aleph_{\omega_1})=\omega_1$

$\text{cf}(\aleph_{\omega_1})=\omega_1$ From here, I quote Brian M. Scott's proof: Suppose that $\langle\alpha_n:n\in\omega\rangle$ is an increasing sequence cofinal in $\omega_{\omega_1}$. For ...
0
votes
1answer
60 views

Let $(\alpha_\xi\mid\xi<\kappa)$ be a sequence such that $\{\alpha_\xi\mid\xi<\kappa\}=\alpha$. Find an increasing subsequence that has limit $\alpha$

Let $\alpha$ be a limit ordinal which is not a cardinal, and $\kappa=|\alpha|$. Then there exists a bijection from $\kappa$ to $\alpha$, or equivalently, a one-to-one sequence $\langle \alpha_\xi \...
1
vote
1answer
72 views

How can we determine $2^\kappa$ for singular $\kappa$ assuming that $2^\lambda=\lambda^+$ whenever $2^{\operatorname{cof}(\lambda)}<\lambda$?

I got an exercise in set theory and can't seem to solve it: If we assume that $2^\lambda = \lambda^+ $ holds for every singular cardinal with $2^{\operatorname{cof}(\lambda)}<\lambda$, then how can ...
0
votes
1answer
43 views

If $\alpha$ is a limit ordinal, then $\operatorname{cf}(\alpha)$ is a limit ordinal [duplicate]

In the textbook Introduction to Set Theory by Hrbacek and Jech, Section 9.2, the authors first introduce the definition of increasing sequence of ordinals: Then they introduce cofinality: My ...
0
votes
0answers
49 views

Is this an equivalent definition of singular cardinal?

In my textbook Introduction to Set Theory, the authors define singular cardinal as follows: I propose an equivalent definition as follows: An infinite cardinal $\kappa$ is called singular if ...
1
vote
1answer
59 views

Jech Lemma 3.10 (cofinalities)

For some reason, this lemma remained elusive in my attempts to find it on the web. I couldn't find it in Hrbacek & Jech either. I understand every part of the proof above except for the last ...
0
votes
1answer
57 views

Jech Lemma 3.7: Why does this follow?

I'm on Jech Chapter 3 (Cardinal Numbers) on the section on cofinalities. I don't understand why the implication in the red rectangle is true. If the aforementioned gamma-sequence was constant and ...
0
votes
0answers
57 views

Is infinite union of finite sets countable?

I wasn't sure of whether infinite union of finite sets is countable? My logic : if $$ A_{i} $$ is a finite subset of any countable set $B$ then:- $$ \bigcup_{i=1}^{\infty }A_{i} =B $$ And we know ...
0
votes
0answers
26 views

Detemine cardinality of $ \{ f: \mathbb{N} \rightarrow [0,1] \}$ [duplicate]

Detemine cardinality of $ \{f \mid f: \mathbb{N} \rightarrow [0,1] \}$. We can show ran$(f) \neq [0,1] $ by a diagonal argument. But I am not sure how to use this to determine the cardinality.
0
votes
1answer
61 views

Almost free modules, PCF theory bound on $2^{\aleph_\omega}$

Why here (Almost Free Modules: Set-theoretic Methods by P.C. Eklof, A.H. Mekler), on the page 181, in the 3rd line there is $$2^{\aleph_\omega}<\aleph_{\omega_2}$$? How the index $2$ in the r.h.s. ...
0
votes
0answers
18 views

Prove that the set $S$ of rational points in the plane $\mathbb R^2$ is denumerable. [duplicate]

Prove that the set $S$ of rational points in the plane $\mathbb R^2$ is denumerable. [A point $p = (x,y) \in \mathbb R^2$ is rational if $x$ and $y$ are rational.]
2
votes
2answers
54 views

“Weak” Ramsey conditions for cardinals

Ok, so these questions just popped into my head and I can't seem to figure it out: Ramsey's theorem tells us that for any $n,r\in\omega$ and any $f:[\omega]^{n}\rightarrow r$, exists an infinite set $...
0
votes
1answer
45 views

Cardinality of nested infinite subsets

Let $A$ and $C$ be infinite sets, with $C \subset A$, and suppose $|A|=|C|$. Now suppose there exists a set $B$ such that $C \subset B \subset A$. Intuitively, $A$ and $B$ should have the same ...
1
vote
0answers
17 views

Cardinality of cartesian products of a series of infinite sets

Let $\hat P(A)=P(A)-\{\emptyset\}$, and $A_0=\mathbb{N}$ and $A_{n+1}=\hat P(A_n)$. I was asked to prove $|A_n|=|A_n\times A_n|$ for any $n\in\mathbb{N}$. I thought about using induction: We have a ...
0
votes
1answer
54 views

What is the cardinality of the following set?

What is the cardinality of the following set: $\{f: \mathbb{R} \rightarrow \{a+b\sqrt[3]4\ | a,b \in \mathbb{Q}\}\}$? Let $S = \{f: \mathbb{R} \rightarrow \{a+b\sqrt[3]4\ | a,b \in \mathbb{Q}\}\}$. ...
1
vote
2answers
35 views

Bijection between $\mathbb{Q}$ and $\mathbb{N}\times\mathbb{Q}$

I want to prove that $|\mathbb{Q}| = |\mathbb{N}\times\mathbb{Q}|$, but I have no idea how to find bijection between these sets. Can you help me?
0
votes
1answer
55 views

Club sets and diagonal intersection

Let $\kappa$ be any regular cardinal, and let $<C_i\mid i<\kappa>$ be a sequence of club sets. Define their diagonal intersection $\Delta C_i$ as follows: $\Delta C_i = \{\alpha<\kappa \...
0
votes
1answer
26 views

How to prove #R + #P = #R

I have already started this. I redefined the Reals as the Reals minus the Positive Integers (to make the two sets disjoint) so that I could prove that #(R - P) + #P = #R. I know that to prove this I ...
0
votes
1answer
81 views

There are arbitrary large singular cardinals $\aleph_\alpha$ such that $\aleph_\alpha=\alpha$

Definitions: Let $(\alpha_\xi \mid \xi < \lambda)$ be a transfinite sequence of ordinals of length $\lambda$. We say that the sequence is increasing if $\alpha_\nu < \alpha_\mu$ whenever $\nu&...
0
votes
1answer
55 views

Why is $\{f\,|\,f\colon A\to\mathbb N\}$ not uncountable?

Let $S=\{\,f\,|\,f\colon A\to\mathbb N\}$, where $A=\{1,2\}$. I thought cardinality of $S$ is $2^{|\mathbb N|}=\aleph_0$. But my friend told that my answer was wrong. Please help me where is I am ...
1
vote
2answers
77 views

Number of ways to split an ordered set

Start with a totally ordered set $A$ of size (cardinality) $N$. What is the size of the set $S$ of subsets of $A$ such that if $y$ is in the subset, any element $x$ in $S$ such that $x < y$ is also ...
1
vote
1answer
30 views

Assuming GCH: if $\mathrm{cf}(\kappa) \leq \lambda < \kappa$, then $\kappa^\lambda = \kappa^+$ (Jech Theorem 5.15)

I am trying to fill in the details for part (ii) of Theorem 5.15 in Jech's Set Theory: Theorem 5.15 If GCH holds and $\kappa$ and $\lambda$ are infinite cardinals, then (i) If $\kappa \leq \...
4
votes
1answer
49 views

What are the properties of countably infinite sets compared to sets of higher cardinality?

When looking at mathematical definitions, there are quite a few cases where we limit certain properties to countably infinite sets (e.g. $\sigma$-Algebras). In some cases we set this limit as we'd ...
0
votes
1answer
49 views

Cardinality bounds on set algebra

Assume two sets $A$ and $B$. Consider the following set of inequalities: $\max \left(\lvert A \rvert, \lvert B\rvert \right) \le \lvert A \cup B\rvert \le \lvert A \rvert + \lvert B \rvert $ $\lvert ...
0
votes
1answer
35 views

How many points are there in the following set? [closed]

Let us consider the following set: $A=\{(x, y, z) \in \Bbb{R}\times\Bbb{R}\times\Bbb{R} : ax+by+c=0,z=0 \},c\neq 0$ and $B=\{(x, y, z) \in \Bbb{R}\times\Bbb{R} \times\Bbb{R} : ax+by=0,z=0\}$. Then ...
4
votes
1answer
38 views

Is the concept of dimension still well defined for non-finite dimensional spaces? [duplicate]

The question is quite simple: if $\mathbb{V}$ is a vector space and $B$ and $B'$ are basis for $\mathbb V$, then do $B$ and $B'$ have the same cardinality? I've tried to answer the question as ...
0
votes
0answers
25 views

Is the denumerable union of denumerable sets denumerable? [duplicate]

I was doing some homework exercises and I'm troubling with this problem. I let $\mathcal{F}:=\{S_1,S_2,\dots,S_n\}$ be any denumerable family, with $S_i$ is a denumerable set for all $i\in\mathbb{N}$, ...
6
votes
2answers
99 views

How to show $2^{ℵ_0} \leq \mathfrak c$ [duplicate]

I want to show $2^{ℵ_0}=\mathfrak c$. I already showed $\mathfrak c \leq 2^{ℵ_0}$ as follows: Each real number is constructed from an integer part and a decimal fraction. The decimal fraction is ...
0
votes
1answer
29 views

Is it true that for any cardinal $\kappa$ that in injects into $\aleph_{\omega}$, $(\kappa)^{+}$ injects into $\aleph_{\omega}$?

Here, $(\kappa)^+$ is the Hartog's Number, the least cardinal $\lambda$ so that there is no injection from $\lambda$ to $\kappa$. My feeling is that this is true, since for any ordinal $\alpha < \...
0
votes
1answer
54 views

What is the cardinality of the set of all functions mapping $\{\sqrt{2},\sqrt{3}, \sqrt{5}, \sqrt{7}\}$ into the rational numbers?

Let $T = \{\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7}\}$ and $\mathcal{S}$ the set of all functions that maps $T$ into $\mathbb{Q}.$ What is the cardinality of $\mathcal{S}$? So far I've been messing with ...
0
votes
0answers
36 views

Prove that $\mathbb{R} \nsim \mathbb{R}^{\mathbb{R}}$. [duplicate]

The task is to prove that $\mathbb{R} \nsim \mathbb{R}^{\mathbb{R}}$. I know how easy it is to prove this once I can use cardinality, but I can't. I need to show that a bijection between those sets ...
0
votes
3answers
28 views

Constructing a Bijection to Demonstrate Countably Infinite Set

I'm looking to prove that $\mathbb{Z} \times \mathbb{Q}$ is countably infinite by constructing a bijection $f: \mathbb{Z} \times \mathbb{Q} \rightarrow \mathbb{N}$. I understand that I can demonstrate ...
0
votes
0answers
77 views

Does $\aleph_0!=\omega$? [duplicate]

More generally, is the order type of some cardinal $\alpha$ equal to $\alpha!$? Related What is $\aleph_0!$? Factorial of Infinite Cardinal factorial of infinite Cardinals